Scanning tunneling microscope study of quantum Hall isospin ferromagnetic states in the zero Landau level in a graphene monolayer

A number of quantum Hall isospin ferromagnetic (QHIFM) states have been predicted in the ``relativistic'' zero Landau level (LL) of the graphene monolayer. However, identification of these high-field broken-symmetry states has mostly relied on macroscopic transport techniques, which lack spatial resolution. Here, we demonstrate a direct approach by imaging the QHIFM states at atomic scale with a scanning tunneling microscope. At half filling of the zero LL ($\ensuremath{\nu}=0$), the system is in a spin unpolarized state and we observe a linear magnetic-field-scaling of valley splitting. The wave functions of the QHIFM states at $\ensuremath{\nu}=0$ are directly imaged at the atomic scale and we observe an interaction-driven density wave featuring a Kekul\'e distortion, which is responsible for the large gap in high magnetic fields. Moreover, our experiment demonstrates that both the valley and spin splittings depend on the filling factor. For example, the spin splitting in the zero LL is dramatically enhanced (in excess of about 200% at maximum) when the Fermi level lies inside spin-polarized states (at $\ensuremath{\nu}=1$ or \ensuremath{-}1), accounting for strong many-body effects.